Cardinal invariants of idealized Miller null sets

Abstract

This paper provides an extensive study of the $\mathcal{I}$-Miller null ideals $M_{\mathcal{I}}$, $\ddot{I}ƒ$-ideals on the Baire space parametrized by ideals $\mathcal{I}$ on countable sets. These $\ddot{I}ƒ$-ideals are associated to the idealized versions of Miller forcing in the same way that the meager ideal is associated to Cohen forcing. We compute the cardinal invariants of $M_{\mathcal{I}}$ for typical examples of Borel ideals $\mathcal{I}$ and show that Cichoń's Maximum can be extended by adding the uniformity and covering numbers of $M_{\mathcal{I}}$ for different ideals $\mathcal{I}$.

66 pages